System and method for magnitude and phase retrieval by path modulation

ABSTRACT

A system includes a transmitter is configured to transmit an electromagnetic signal through a sample cell (including a sample medium) to a receiver, which is configured to receive the electromagnetic signal and another electromagnetic signal for mixing therewith. Propagation paths of the signals to the transmitter and receiver include a first propagation path of the electromagnetic signal to the transmitter, and a second propagation path of the other electromagnetic signal to the receiver. The arrangement, which is located along either or each of the propagation paths of signals to the transmitter and receiver, is configured to alter the length of a respective propagation path. And the processor configured to recover an amplitude and phase of the transmitted electromagnetic signal, and calculate a complex index of refraction of the sample medium as a function of the amplitude and phase of the transmitted electromagnetic signal.

FIELD OF THE INVENTION

Exemplary embodiments of present invention generally relate to systemsand methods of propagating electromagnetic signals and, moreparticularly, systems and methods of magnitude and phase retrieval bypath modulation.

BACKGROUND OF THE INVENTION

Spectrometry using continuous wave (CW) tunable sources with narrowspectral linewidth and long coherence lengths has well-known advantagesassociated with high spectral contrast, frequency selectivity andexcellent sensitivity. Scanning CW terahertz (THz) spectrometers are aprime example of this technology. In such systems, phase stability inthe transmitter-to-receiver demodulation processing may be required toobtain an accurate measurement of the transmitted electric-fieldintensity and to characterize any resulting absorption losses fromsamples in the spectrometer. However, spurious thermal and mechanicaldisturbances may undesirably generate variations in path length thatmodulate the received field intensity. It would therefore be desirableto design a system and method of controlling and modulating the pathlength to improve demodulation signal to noise ratio and stability.

SUMMARY OF THE INVENTION

In light of the foregoing background, embodiments of the presentinvention provide an improved system and method of magnitude and phaseretrieval by path modulation. According to one aspect of the presentinvention, the system includes a transmitter configured to transmit anelectromagnetic signal through a sample cell (including a sample mediumand a base medium) to a receiver at each of one or more selectablefrequencies, where the receiver is configured to receive theelectromagnetic signal and another electromagnetic signal for mixingtherewith. The system also includes an arrangement located along eitherof first or second propagation paths of signals to the transmitter orreceiver, respectively, or along each of the first and secondpropagation paths, for altering the length of respective propagationpath(s).

The system also includes a processor configured to recover an amplitudeand phase of the transmitted electromagnetic signal, and configured tocalculate a complex index of refraction of the sample medium as afunction of the amplitude and phase of the transmitted electromagneticsignal. The processor may be configured to receive a sequence of samplesof the received electromagnetic signal, and Discrete FourierTransformation process the sequence of samples. In addition, the systemmay include a modulator configured to modulate the electromagneticsignal transmitted by the transmitter, where the modulator may beconfigured to modulate the electromagnetic signal at a frequency (e.g.,ω_(m)), which may be above the 1/f noise region of the receiver.

The arrangement is configured to alter the length of a respectivepropagation path such that the difference of the lengths of the firstand second propagation paths is altered at a pre-selected rate duringtransmission of the electromagnetic signal from the transmitter to thereceiver, and receipt of the electromagnetic signal and the otherelectromagnetic signal at the receiver. The arrangement may comprise apair of arrangements each of which is located along a respective one ofthe first and second propagation paths. To effectuate an increase in thedifference in the lengths of the first and second propagation paths atthe pre-selected rate, one of the arrangements may be configured toincrease the length of one of the propagation paths, while the other ofthe arrangements may be configured to decrease the length of the otherof the propagation paths. The arrangement may include, for example, aspool and actuator. In such instances, an optical fiber propagation pathmay be wound about the spool, and the actuator may be configured toalter the diameter of the spool, and thereby alter the length of therespective propagation path.

The processor being configured to calculate a complex index ofrefraction of the sample medium may include being configured tocalculate a real part n_(s) and an imaginary part k of the complex indexof refraction. The real part of the complex index of refraction may becalculated as a function of the recovered phase φ_(SAMPLE) of thetransmitted electromagnetic signal, and as a function of a recoveredphase φ_(REF) of an electromagnetic signal transmitted through thesample cell including the base medium but without the sample medium.Similarly, the imaginary part of the complex index of refraction may becalculated as a function of the recovered amplitude E_(s) of thetransmitted electromagnetic signal, and as a function of a recoveredamplitude E_(o) of an electromagnetic signal transmitted through thesample cell including the base medium but without the sample medium. Forexample, the real part of the complex index of refraction may becalculated according to the following:

$n_{s} = {{\left( {\varphi_{REF} - \varphi_{SAMPLE}} \right) \times \frac{\lambda}{2\; \pi \times L_{S}}} + n_{o}}$

where λ (e.g., λ_(THz)) represents a wavelength of the transmittedelectromagnetic signal, L_(s) represents a propagation path lengththrough the sample medium, and n_(o) represents the free-space index ofrefraction. And the imaginary part of the complex index of refractionmay be calculated according to the following:

$k = {{- \frac{\lambda}{2\; \pi \; L_{s}}}{\ln \left( \frac{E_{s}}{E_{o}} \right)}}$

The pre-selected rate may comprise a rate selected as a function of thefrequency at which the electromagnetic signal is transmitted. Moreparticularly, the pre-selected rate may comprise a rate selected to spanone or more wavelengths of the electromagnetic signal transmitted at arespective frequency over a dwell time. In one instance, for example,the pre-selected rate may comprise a rate ω_(FS) selected to effectuatea path length modulation at a frequency:

$\omega_{FS} = {\frac{2\pi}{\lambda}n_{F}S_{F}}$

In the preceding, n_(F) represents the index of refraction of apropagating medium of the propagation paths, and S_(F) represents thepre-selected rate. In this regard, the processor may be configured toDFT process the sequence of samples at a first frequency of interestω_(m)−ω_(FS), and DFT process the sequence of samples at a secondfrequency of interest ω_(m)+ω_(FS). And as the pre-selected rate maycomprise a rate selected as a function of frequency, the pre-selectedrate may comprise a rate selected for each of the one or more selectablefrequencies, where the rate selected for one of the frequencies maydiffer from the rate selected for another of the frequencies.

According to other aspects of the present invention, a method ofmagnitude and phase retrieval by path modulation is provided. Exemplaryembodiments of the present invention therefore provide an improvedsystem and method of magnitude and phase retrieval by path modulation.As indicated above, and explained below, exemplary embodiments of thepresent invention may solve problems identified by prior techniques andprovide additional advantages.

BRIEF DESCRIPTION OF THE DRAWINGS

Having thus described the invention in general terms, reference will nowbe made to the accompanying drawings, which are not necessarily drawn toscale, and wherein:

FIG. 1 is a schematic block diagram of a spectrometer system inaccordance with one exemplary embodiment of the present invention;

FIGS. 2 and 3 are flowcharts illustrating various steps in methods ofsweeping a spectrometer system through a frequency spectrum, accordingto exemplary embodiments of the present invention;

FIG. 4 is a graph illustrating the measured noise density spectrum of aphotomixer receiver, according to exemplary embodiments of the presentinvention; and

FIG. 5 illustrates spectral diagrams illustrating frequency downconversion in the receiver of exemplary embodiments of the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention now will be described more fully hereinafter withreference to the accompanying drawings, in which preferred embodimentsof the invention are shown. This invention may, however, be embodied inmany different forms and should not be construed as limited to theembodiments set forth herein; rather, these embodiments are provided sothat this disclosure will be thorough and complete, and will fullyconvey the scope of the invention to those skilled in the art. In thisregard, reference may be made herein to a number of mathematical ornumerical expressions that may be related by equality. It should beunderstood, however, that this equality may refer to an absolute orapproximate equality, such that exemplary embodiments of the presentinvention may account for variations that may occur in the system andmethod, such as those due to engineering tolerances. Further, although anumber of variables may be reflected by mathematical symbols includingsubscripts at various instances, it should be understood that thesesymbols and subscripts are presented solely for illustrative purposes,and should not be construed as limiting the scope of the invention. Likenumbers refer to like elements throughout.

FIGS. 1 and 2 illustrate a spectrometer system and method that maybenefit from exemplary embodiments of the present invention (“exemplary”as used herein referring to “serving as an example, instance orillustration”). It should be understood, however, that the spectrometersystem and method illustrated and hereinafter described are merelyillustrative of one type of system and method that may benefit fromexemplary embodiments of the present invention and, therefore, shouldnot be taken to limit the scope of the present invention. In thisregard, while several embodiments of the spectrometer system and methodare illustrated and will be hereinafter described for purposes ofexample, other types of systems and methods of propagatingelectromagnetic signals may readily employ the present invention.Moreover, the system and method of the present invention will beprimarily described in conjunction with signals in the THz (or mmW)region of the electromagnetic spectrum. But the system and method ofembodiments of the present invention may be utilized in conjunction witha variety of other applications, both within and outside the THz regionof the electromagnetic spectrum.

As shown, a spectrometer system 10 of one exemplary embodiment of thepresent invention includes a transmitter 12 configured to transmit abeam of coherent radiation (electromagnetic wave) at a given frequency.The transmitter can comprise any of a number of different transmittersknown to those skilled in the art. In one exemplary embodiment, forexample, the transmitter comprises a photomixer transmitter. In suchinstances, the transmitter includes a high-speed photoconductive diode(i.e., photomixer), which may be pumped with two laser sources 14 a, 14b via a beam combiner/splitter 16 and an optically coupled first opticalpath 18 (e.g., optical fiber). In this regard, the laser sources may beconfigured to emit signals with electric fields having offsettingfrequencies at ω₁ and ω₂ (i.e., E_(ω1) and E_(ω2)), which at thephotomixer transmitter may be represented as follows:

E _(ω1) =E ₁ cos(ω₁ t+φ _(1T))  (1)

E _(ω2) =E ₂ cos(ω₂ t+φ _(2T))  (2)

where E₁ and E₂ represent the electric-field amplitudes of the beamsfrom the first and second sources, respectively; and φ_(1T) and φ_(2T)represent phase constants introduced by virtue of propagation of thebeams through the first optical path. Also note that frequencies ω₁ andω₂ may be expressed as angular frequencies, or as corresponding temporalfrequencies (f=ω/2π).

The inherently quadratic nature of the cross-gap absorption creates adifference (i.e., transmission) frequency (i.e., ω₂−ω₁) in thephotocurrent induced in the diode of the transmitter 12, where thecorresponding electric field may be represented as follows:

E _(T)=η_(T) E ₁ E ₂ cos(ω_(THz) t+φ _(12T))  (3)

where η_(T) represents the photomixer transmitter conversion efficiency,ω_(THz)=ω₂−ω₁, and φ_(12T)=φ_(2T)−φ_(1T). The transmitter 12 may becoupled to a transmitter bias modulator 20 including a voltage source 22configured to generate a sinusoidal modulated voltage with which thephotomixer of the transmitter may be biased, the modulator producing anelectric field E_(M)=V_(m) cos(ω_(m)t), although it should be understoodthat the system need not frequency modulate (at frequency ω_(m)) thesignal. By locating the photomixer at the driving point of an antenna,such as a spiral, dipole or slot antenna, the difference-frequencycurrent is converted to difference-frequency photons. The result is ahighly-tunable, continuous-wave (CW), highly-coherent source ofradiation contained in a single (quasi-Gaussian) spatial mode, andhaving a transmitted electric field E_(TM). This transmitted electricfield may be represented as the product of E_(T) and E_(M), as follows:

$\begin{matrix}{E_{TM} = {V_{m}{\cos \left( {\omega_{m}t} \right)}\eta_{T}E_{1}E_{2}{\cos \left( {{\omega_{THz}t} + \varphi_{T}} \right)}}} & (4) \\{E_{TM} = {\frac{V_{m}\eta_{T}E_{1}E_{2}}{2}\begin{bmatrix}{{\cos \left( {{\left( {\omega_{THz} + \omega_{m}} \right)t} + \varphi_{T}} \right)} +} \\{\cos \left( {{\left( {\omega_{THz} - \omega_{m}} \right)t} + \varphi_{T}} \right)}\end{bmatrix}}} & (5)\end{matrix}$

In equations (4) and (5), φ_(T) represents the sum of φ_(12T) and somephase delay related to the photomixer and antenna transfer function,which may be significant and may cause large signal variability whendetected at a receiver. For more information on such a transmitter, seeU.S. Pat. No. 6,348,683 entitled: Quasi-Optical Transceiver Having anAntenna with Time Varying Voltage, issued Feb. 19, 2002.

Thus, the method of one embodiment includes selecting a transmissionfrequency, thereafter transmitting a beam of radiation (i.e., sourcebeam) at that frequency from the transmitter 12, as shown in blocks 42and 48 of FIG. 2. The transmission frequency can be selected in any of anumber of different manners. To detect a sample based upon a measuredabsorption signature, however, the transmission frequency may betypically selected within a range of frequencies over which theabsorption signature is defined. In a photomixer transmitter, then, thephotomixer can be pumped with tunable laser sources at a frequency ω₂,and a frequency ω₁ that are selected to thereby select the difference,or transmission, frequency (i.e., ω₂−ω₁).

The beam of radiation from the transmitter 12 may pass through acollimating lens 24 to produce a collimated beam of radiation. The beammay then pass through a sample cell 26 that may be bounded by reflectors26 a and 26 b through which the beam passes, and that may include asample medium to be analyzed and a base medium, such as ambient air. Aswill be appreciated, the sample and base medium can have any of a numberof different forms through which the beam of radiation is at leastpartially transmissive. For example, the sample and base medium cancomprise a solid, liquid, gas, plasma or aerosol. More particularly, invarious advantageous embodiments, the base medium of ambient air may bein gas form, while a sample may be in gas or aerosol form.

As the beam of radiation passes through the sample cell 26, the sampleand base medium in the sample cell absorb at least a portion of thebeam, or more particularly at least a portion of the electric field ofthe beam. The remaining, unabsorbed portion of the beam of radiation(i.e., received signal) then exits the sample cell. The sample signalthen propagates to a focusing lens 28, from which the focused signal ispicked up or otherwise received by a receiver 30. This received signalE_(RP), which may include an additional phase delay added duringpropagation of the signal from the transmitter 12 to the receiver, maybe represented as follows:

$\begin{matrix}{E_{RP} = {\frac{V_{m}\eta_{T}E_{1}E_{2}}{2} \times ^{{- 2}\pi \; k\frac{L}{\lambda_{THz}}} \times \begin{bmatrix}{{\cos \begin{pmatrix}{{\left( {\omega_{THz} + \omega_{m}} \right)t} +} \\{\varphi_{T} - {2\; \pi \frac{n \times L}{\lambda_{THz}}}}\end{pmatrix}} +} \\{\cos \begin{pmatrix}{{\left( {\omega_{THz} - \omega_{m}} \right)t} +} \\{\varphi_{T} - {2\; \pi \frac{n \times L}{\lambda_{THz}}}}\end{pmatrix}}\end{bmatrix}}} & (6)\end{matrix}$

where L represents the propagation distance between the transmitter andreceiver, λ_(THz) represents the wavelengths of the signal at thefrequency ω_(THz), n and k represent the real part and the imaginarypart (extinction coefficient), respectively, of the complex index ofrefraction of the sample in the path length L.

The receiver obtains a measurement representative of the receivedelectric field E_(RP), as shown in block 50 of FIG. 2. Similar to thetransmitter 12, the receiver may comprise an electric-field detectorsuch as a photomixer receiver (homodyne receiver). The photomixerreceiver may include an antenna configured to receive the electric fieldand generate a corresponding voltage in response thereto, which may bedirected to a high-speed photoconductor. The photoconductor is alsoelectrically coupled to a second optical path 32 for pumping thephotoconductor with beams from the same two laser sources 14 a, 14 bpumping the photomixer transmitter 12. In this regard, the beamcombiner/splitter 16 may separate each of the signals from the lasersources into the aforementioned first optical path 18, as well asanother, second optical path (e.g., optical fiber) for pumping thereceiver photomixer. These signals, then, may modulate a conductance ofthe photomixer as described by the following:

G _(RP)=η_(R) E ₁ E ₂ cos(ω_(THz) t+φ _(12R))  (7)

where η_(R) represents the photomixer receiver conversion efficiency,and φ_(12R)=φ_(2R)−φ_(1R), φ_(1R) and φ_(2R) representing phaseconstants introduced by virtue of propagation of the beams through thesecond optical path.

The voltage generated by the receiver antenna may be applied to thephotomixer active material, and produce a current through the modulatedconductance that is the product of equations (6) and (7). The differencefrequency result of the product is the down-converted signal currentI_(Down), which may be represented as follows:

$\begin{matrix}{{I_{Down} = {\frac{E_{RPmp}G_{RPmp}}{2} \times ^{{- 2}\pi \; k\frac{L}{\lambda_{THz}}} \times \begin{bmatrix}{{\cos \left( {{\omega_{m}t} + \varphi_{T} - {2\; \pi \frac{n \times L}{\lambda_{THz}}} - \varphi_{12R}} \right)} +} \\{\cos \left( {{\omega_{m}t} - \varphi_{T} + {2\; \pi \frac{n \times L}{\lambda_{THz}}} + \varphi_{12R}} \right)}\end{bmatrix}}}\mspace{79mu} {{where},\mspace{79mu} {E_{RPmp} = {{\frac{V_{m}\eta_{T}E_{1}E_{2}}{2}\mspace{14mu} {and}\mspace{14mu} G_{RPmp}} = {\eta_{R}E_{1}E_{2}}}}}} & (8)\end{matrix}$

A corresponding down-converted electric-field (or signal) E_(R), then,may be calculated as follows:

E_(R)=I_(Down)R_(Load)  (9)

where R_(Load) represents the receiver 30 electronic load resistance.For more information on such a receiver, see the aforementioned '683patent.

The down-converted signal current I_(Down) and/or electric-field (orsignal) E_(R) may be applied to receiver signal conditioning circuitry34 including, for example, an anti-aliasing filter 36. The output of thesignal conditioning circuitry may then be input to a processor 38, suchas for performing digital signal processing operations thereon. In thisregard, the processor can comprise any of a number of differentprocessing devices capable of operating in accordance with exemplaryembodiments of the present invention. For example, the processor cancomprise a computer (e.g., personal computer, laptop computer, servercomputer, workstation computer), microprocessor, coprocessor,controller, a specialized digital signal processor and/or various otherprocessing devices including integrated circuits such as an ASIC(application specific integrated circuit), FPGA (field programmable gatearray) or the like.

If the spectrometer system 10 frequency modulates (at frequency ω_(m))the signal, the signal processing operations performed by the processor38 may include recovering the amplitude of the down-converted signalE_(R) such as by an analog-to-digital converter (A/D) 39 direct samplingof the signal at the modulating frequency ω_(m), and the processorDiscrete Fourier Transformation (DFT) processing of the sampled data.Alternatively, for example, the spectrometer system may further includea synchronous demodulator such as a lock-in amplifier (not shown) forfurther processing the down-converted signal E_(R). In this regard, sucha synchronous demodulator may include a local oscillator operating atthe modulating frequency ω_(m) to thereby recover the amplitude of thedown-converted signal.

In operation as a spectrometer, the system 10 scans through a number oftransmission frequencies in a range of frequencies, such as by pumpingthe photomixers of the transmitter 12 and receiver 30 with tunable lasersources at frequency ω₂, and frequency ω₁ that are scanned through anumber of frequencies, as shown in blocks 54 and 56 of FIG. 2. For eachtransmission frequency in the range of frequency, and thus each beam ofradiation having a different transmission frequency, the processor 38may measure the amplitude and/or phase of the down-converted signalcurrent I_(Down). The resulting collection of transmissions amplitudesand/or phases, and associated transmission frequencies, may define ameasured absorption or dispersion signature for the sample in the samplecell 26, from which the sample may be identified, as shown in block 58of FIG. 2.

As explained in the background section, in certain optical transmissionsystems (e.g., spectrometer systems), spurious thermal and mechanicaldisturbances may undesirably generate variations in path length thatmodulate the received field phase and down-converted amplitude.Exemplary embodiments of the present invention therefore provide anapparatus and method of modulating the path length of either or both ofthe first or second optical paths 18, 32 to at least partially rejectpath length instabilities that may produce transmitted signal amplitudeerrors. The applied path length modulation may allow recovery of thetransmitted signal amplitude during a relatively short dwell time of themeasurement of each spectral point (selected frequency) of the scanningspectrometer. In this regard, phase modulation by spurious pathvariations may be mitigated through the intentional path stretching thatspans multiple waves of the transmitted frequency. And with multiplewave modulation, the full amplitude of the transmitted signal may beobserved and errors in signal amplitude may be suppressed.

According to exemplary embodiments, the spectrometer system 10 furtherincludes a path length modulation arrangement 40 along either the firstoptical path or the second optical path, or as shown, or along each ofthe first and second optical paths. Exemplary embodiments may apply pathlength modulation to either or both of the optical paths, and in equalor differing amounts, to thereby effectuate a total system path lengthstretch. In this regard, when simultaneously applying modulation to bothof the optical paths, the resulting system path modulation or stretchmay correspond to the difference of the modulation applied to the firstand second optical paths, and may require contraction (decreasing thelength) of one of the paths as the other path is stretched (increasingthe length).

The path length modulation arrangement 40 may comprise any of a numberof apparatuses configured to dynamically stretch or contract an opticalpath length. In one exemplary embodiment in which an optical pathincludes an optical fiber, the path length modulation arrangement maycomprise a spool about which the fiber may be wound, and an actuator(e.g., piezoelectric actuator) coupled to the spool configured tostretch the diameter of the spool and thus the length of the fiber woundthereabout. In such instances, contraction of the optical fiber may beeffectuated by reducing a previously-applied stretch to the spool andthus the fiber.

According to exemplary embodiments of the present invention, then,before the laser sources 14 a, 14 b pump the photomixer transmitter 12to thereby transmit a beam of radiation at a selected frequency (seeFIG. 2, block 48), a path length rate scale factor S_(F) may beselected, such as by the processor 38, as shown in block 46 of FIG. 3.The path length rate scale factor represents the rate of applying asystem stretch (stretch of one or both optical paths, or stretch of onepath coupled with contraction of the other path) during the dwell timeat each frequency sample point of the scanned spectrum (i.e., the amountof time the system operates at each frequency sample point before movingto the next point).

The path length rate scale may be selected in any of a number ofdifferent manners to effectuate a desired path length modulation, suchas in a manner so as to span one or more waves of the pump signal (atthe difference frequency) within the optical paths 18, 32 over the dwelltime at each frequency sample. More particularly, for example, the pathlength rate scale may be selected as an integer multiple of the periodof the pump signal, such as in accordance with the following:

$\begin{matrix}{S_{F} = \frac{a\; \lambda_{THz}}{D}} & (10)\end{matrix}$

where a represents a selectable integer multiple (e.g., 3), λ_(THZ)represents the wavelength of the pump signal at the differencefrequency, and D represents the dwell time (e.g., 0.03 sec.). Written interms of the difference frequency f_(THz), the path length rate scalemay be selected as follows:

$\begin{matrix}{S_{F} = {\frac{a}{D}\frac{c}{f_{THz}n_{F}}}} & (11)\end{matrix}$

where n_(F) represents the index of refraction of the propagating mediumof the optical path (e.g., approximately 1.5 for optical fiber).Consider for example, an instance in which a=3, D=0.03 s, f_(THz)=650GHz, and n_(F)=1.5. In such an instance, given c=3×10⁸ m/s, the pathlength rate scale factor S_(F) may be selected as approximately 30.77mm/s.

As relatively low frequencies of the path length modulation may resultin increased noise in the spectrometer system 10, before, as or afterthe path length rate scale factor is selected, a transmitter modulatingfrequency ω_(m) may be selected so as to elevate the signal carrierabove the 1/f noise region of the receiver electronics, as shown inblock 44. This selection of the modulating frequency may permit thesystem to at least partially avoid increased noise at relatively lowfrequencies of the path length modulation. The transmitter modulatingfrequency may be selected in a number of different manners, such as fromanalysis of a measured noise density spectrum of the receiver. Oneexample of a measured noise density spectrum is shown in the graph ofFIG. 4. As shown, the 1/f noise region of the receiver electronics is atapproximately 1 kHz. And from this exemplary noise density spectrum, itmay be shown that a transmitter modulation frequency ω_(m) at or above10 kHz may be needed to at least partially avoid excess 1/f noise.

Having selected the path length rate scale factor S_(F) and transmittermodulating frequency ω_(m), the method may proceed similar to before,including transmitting a beam of radiation (i.e., source beam) at aselected transmission frequency, as shown in block 48 of FIG. 3. As thebeam of radiation is transmitted during the dwell time of the selectedtransmission frequency, the processor 38 may control the path lengthmodulation arrangement(s) 40 (or more particularly, for example, theactuator(s) of the arrangements) to stretch and/or contract the firstoptical path 18 and/or the second optical path 32 to effectuate a totalsystem path length stretch. In such an instance, the emitted signalsE_(ω1) and E_(ω2) may be represented as follows:

E _(ω1) =E ₁ cos(ω₁ t+ω _(FS) t+φ _(1T))  (12)

E _(ω2) =E ₂ cos(ω₂ t+ω _(FS) t+φ _(2T))  (13)

The difference (i.e., transmission) frequency (i.e., ω₂−ω₁) in thephotocurrent induced in the diode of the transmitter 12, then, may havea corresponding electric field:

E _(T)=η_(T) E ₁ E ₂ cos((ω_(THz)+ω_(FS))t+φ ₁₂)  (14)

where ω_(FS) represents the path length modulation frequency at thedifference frequency, which may be represented as follows:

$\begin{matrix}{\omega_{FS} = {\frac{2\; \pi}{\lambda_{THz}}n_{F}S_{F}}} & (15)\end{matrix}$

Similar to before, the transmitter 12 may be coupled to a transmitterbias modulator 20 including a voltage source 22 configured to generate asinusoidal modulated voltage with which the photomixer of thetransmitter may be biased, the modulator producing an electric fieldE_(M)=V_(m) cos(ω_(m)t). The transmitted electric field, then, may berepresented as the product of E_(T) and E_(M)=V_(m) cos(ω_(m)t), asfollows:

$\begin{matrix}{E_{TM} = {V_{m}{\cos \left( {\omega_{m}t} \right)}\eta_{T}E_{1}E_{2}{\cos \left( {{\left( {{\omega_{THz}t} + \omega_{FS}} \right)t} + \varphi_{T}} \right)}}} & (16) \\{E_{TM} = {\frac{V_{m}\eta_{T}E_{1}E_{2}}{2}\begin{bmatrix}{{\cos \left( {{\left( {\omega_{THz} + \omega_{FS} + \omega_{m}} \right)t} + \varphi_{T}} \right)} +} \\{\cos \left( {{\left( {\omega_{THz} + \omega_{FS} - \omega_{m}} \right)t} + \varphi_{T}} \right)}\end{bmatrix}}} & (17)\end{matrix}$

The beam of radiation from the transmitter may, as before, pass throughthe collimating lens 24 and sample cell 26. A portion of the beam ofradiation may exit the sample cell, pass through the focusing lens 28,and be picked up or otherwise received by the receiver 30, as shown atblock 50; and from the received signal, the complex index of refractionof the sample may be calculated, as explained further below and as shownat block 52. This received signal E_(RP) may be represented as follows:

$\begin{matrix}{E_{RP} = {\frac{V_{m}\eta_{T}E_{1}E_{2}}{2} \times ^{{- 2}\pi \; k\frac{L}{\lambda_{THz}}} \times \left\lbrack \begin{matrix}{{\cos \left( {{\left( {\omega_{THz} + \omega_{FS} + \omega_{m}} \right)t} + \varphi_{T} - {2\; \pi \frac{n \times L}{\lambda_{THz}}}} \right)} +} \\{\cos \left( {{\left( {\omega_{THz} + \omega_{FS} - \omega_{m}} \right)t} + \varphi_{T} - {2\; \pi \frac{n \times L}{\lambda_{THz}}}} \right)}\end{matrix} \right\rbrack}} & (18)\end{matrix}$

The receiver 30 may receive the electric field and generate acorresponding voltage in response thereto. The voltage generated by thereceiver may be applied to the photomixer active material, and produce acurrent through the modulated conductance that is the product ofequations (18) and (7). The difference frequency result of the productis the down-converted signal current I_(Down), which may be representedas follows:

$\begin{matrix}{I_{Down} = {\frac{E_{RPmp}G_{RPmp}}{2} \times ^{{- 2}\pi \; k\frac{L}{\lambda_{THz}}} \times \left\lbrack \begin{matrix}{{\cos \left( {{\left( {\omega_{m} + \omega_{FS}} \right)t} + \varphi_{T} - {2\; \pi \frac{n \times L}{\lambda_{THz}}} - \varphi_{12R}} \right)} +} \\{\cos \left( {{\left( {\omega_{m} - \omega_{FS}} \right)t} - \varphi_{T} + {2\; \pi \frac{n \times L}{\lambda_{THz}}} - \varphi_{12R}} \right)}\end{matrix} \right\rbrack}} & (19)\end{matrix}$

The corresponding down-converted electric-field (or signal) E_(R), then,may be calculated according to the following:

$\begin{matrix}{E_{R} = {E_{o} \times ^{{- 2}\pi \; k\frac{L_{s}}{\lambda_{THz}}} \times \begin{bmatrix}{{\cos \begin{pmatrix}{{\left( {\omega_{m} + \omega_{FS}} \right)t} + \varphi -} \\{2\; \pi \frac{{n_{o} \times L_{R}} + {n_{s} \times L_{s}}}{\lambda_{THz}}}\end{pmatrix}} +} \\{\cos \begin{pmatrix}{{\left( {\omega_{m} - \omega_{FS}} \right)t} - \varphi +} \\{2\; \pi \frac{{n_{o} \times L_{R}} + {n_{s} \times L_{s}}}{\lambda_{THz}}}\end{pmatrix}}\end{bmatrix}}} & (20)\end{matrix}$

and may be simplified as follows:

E _(R) =E _(o)[cos((ω_(m)+ω_(FS))t+φ)+cos((ω_(m)−ω_(FS))t−φ)]  (21)

In the preceding equations, L_(R) represents the propagation path lengthin the reference or free-space medium with free-space index ofrefraction n_(o)(=1), L_(s) represents the sample thickness of or apropagation path length through the sample medium with index ofrefraction ñ≡n_(s)+ik (n_(s) and k respectively representing the realand imaginary parts of the complex index ñ), and E_(o) and φ may berepresented as follows:

$E_{o} = {\frac{1}{2}E_{RPmp}G_{RPmp}R_{Load}}$$\varphi = {\varphi_{T} - {2\; \pi \frac{{n_{0} \times L_{R}} + {n_{s} \times L_{s}}}{\lambda_{THz}}} - \varphi_{12R}}$

where E_(o) represents a signal amplitude with no absorption (i.e.,k=0). This result is the mixing product of the receiver photomixer asillustrated in the spectral diagrams of FIG. 5.

Also as before, the down-converted signal current I_(Down) and/orelectric-field (or signal) E_(R) may be applied to receiver signalconditioning circuitry 34 and then input to a processor 38, and mayinclude recovery of the amplitude of the down-converted signal E_(R). Inequations (20) and (21), the constant phase term φ may vary with pathlength drift as a function of temperature and mechanical disturbances.By performing path length modulation according to exemplary embodimentsof the present invention, the received signal amplitude may be extractedat much higher frequencies than any path drift affecting signal phase.And from the received signal amplitude and phases, the complex index ofrefraction of the sample may be calculated.

More particularly, according to one exemplary embodiment of the presentinvention, the amplitude of the down-converted signal E_(R) may berecovered by an analog-to-digital converter (A/D) 39 direct sampling ofthe signal E_(R) at the modulating frequency ω_(m), and the processorDiscrete Fourier Transformation (DFT) processing of the sampled data. Inthis regard, the sequence of sampled values of the signal output fromthe A/D may be represented as follows:

E _(b) =E _(R)(n×τ _(s))  (22)

where b represents the index of the sampled sequence b=0, 1, 2, 3, . . .B−1, B represents the number of samples, and τ_(s) represents a samplingtime interval (in seconds) that may be selected to ensure no aliasing atφ_(FS)+φ_(m).

The sequence of sampled values E_(b) may be received by the processor38, which may then perform DFT processing of the sampled values.Applying Euler's formula to the DFT X(f) of a complex sequencex(n×τ_(s)), the DFT may be represented as:

$\begin{matrix}{{X(f)} = {\sum\limits_{b = 0}^{B - 1}{{x\left( {b \times \tau_{s}} \right)} \times \left( {{\cos \left( {2\pi \; {fb}\; \tau_{s}} \right)} + {i\; {\sin \left( {2\pi \; {fb}\; \tau_{s}} \right)}}} \right)}}} & (23)\end{matrix}$

where f represents the frequency of the sinusoid at which the DFT isevaluated (in Hz). Presuming a frequency of interest of ω_(m)−ω_(FS) andsetting the frequency 2πf=ω_(m)−ω_(FS), and additionally applying anormalization of 2/N, the DFT of the sampled values E_(b) may berepresented as:

$\begin{matrix}\begin{matrix}{{X(f)} = X_{\omega_{m} - \omega_{FS}}} \\{= {\frac{2}{B}{\sum\limits_{b = 0}^{B - 1}{E_{b} \times \left( {{\cos \left( {\left( {\omega_{m} - \omega_{FS}} \right)b\; \tau_{s}} \right)} + {i\; {\sin \left( {\left( {\omega_{m} - \omega_{FS}} \right)b\; \tau_{s}} \right)}}} \right)}}}}\end{matrix} & (24)\end{matrix}$

The sequence of functions may be equally applied to the frequency ofinterest ω_(m)−ω_(FS) (setting 2πf=ω_(m)−ω_(FS)) to yield:

$\begin{matrix}\begin{matrix}{{X(f)} = X_{\omega_{m} - \omega_{FS}}} \\{= {\frac{2}{B}{\sum\limits_{b = 0}^{B - 1}{E_{b} \times \left( {{\cos \left( {\left( {\omega_{m} - \omega_{FS}} \right)b\; \tau_{s}} \right)} + {i\; {\sin \left( {\left( {\omega_{m} - \omega_{FS}} \right)b\; \tau_{s}} \right)}}} \right)}}}}\end{matrix} & (25)\end{matrix}$

The processor 38 may calculate the amplitude and phase only for thefrequency components of interest (ω_(m)−ω_(FS)) and (ω_(m)+ω_(FS)) as afunction of the DFTs equations (24) and (25). These results may becomplex values whose magnitudes are equal and phases are of oppositesign as indicated by equations (20) and (21). The reported, transmittedamplitude may then be calculated as the average of the above twomagnitudes. Phase information may be used to derive the complex index ofthe medium in the propagation path. Notably, the resulting processingsignal bandwidth may be limited by the transform of the rectangularwindow whose width may be the total sample time, i.e., B×τ_(s).Windowing functions common to Fast Fourier Transform processing may beapplied to manage the sidelobes and width of the processing passband.

The phase information that leads to the real part of the index ofrefraction (i.e., n or n_(s)) may be derived from equations (24) and(25) as follows:

$\begin{matrix}{{\varphi - {2\pi \frac{n \times L}{\lambda_{THz}}}} = {\arctan\left( \frac{\sum\limits_{B = 0}^{B - 1}{E_{b} \times {\sin \left( {\left( {\omega_{m} + \omega_{FS}} \right)b\; \tau_{s}} \right)}}}{\sum\limits_{B = 0}^{N - 1}{E_{b} \times {\cos \left( {\left( {\omega_{m} + \omega_{FS}} \right)b\; \tau_{s}} \right)}}} \right)}} & (26)\end{matrix}$

Performing a similar computation using the lower sideband((ω_(m)−ω_(FS)) components results in the same phase value with theopposite sign, i.e., negated. The complex index of refraction of thesample in the propagation path ñ may be derived from the measuredamplitude and phase expressed in equation (20) by first obtaining orotherwise retrieving (e.g., from memory embodied within or otherwiseassociated with the processor 38) reference measurements of theamplitude (E_(o)) and phase (φ_(REF)) for a transmitted signal passingthrough the sample cell 26 with the base medium (e.g., ambient air) butwithout the sample medium. Then, with the sample medium having beeninserted into the cell, and sample measurements of the amplitude (E_(s))and phase (φ_(SAMPLE)) may be obtained for a transmitted signal passingthrough the sample cell including the sample and base mediums. Thereference and sample phases may be represented as follows:

$\begin{matrix}{\varphi_{REF} = {\varphi - {2\pi \frac{n_{o} \times L_{R}}{\lambda_{THz}}}}} & (27) \\{\varphi_{SAMPLE} = {\varphi - {2\pi \frac{{n_{o} \times \left( {L_{R} - L_{s}} \right)} + {n_{s} \times L_{s}}}{\lambda_{Thz}}}}} & (28)\end{matrix}$

The reference and sample amplitudes may be represented as in equations(24) and (25), and may be averaged over the frequency components ofinterest (ω_(m)−ω_(FS)) and (ω_(m)+ω_(FS)) to yield:

$\begin{matrix}{E_{o} = \left. \frac{{X_{\omega_{m} - \omega_{FS}}} + {X_{\omega_{m} + \omega_{FS}}}}{2} \right|_{REF}} & (29) \\{E_{s} = \left. \frac{{X_{\omega_{m} - \omega_{FS}}} + {X_{\omega_{m} + \omega_{FS}}}}{2} \right|_{SAMPLE}} & (30)\end{matrix}$

From the preceding measurements and their representations, the real partof the index of refraction of the sample may be obtained as follows:

$\begin{matrix}{n_{s} = {{\left( {\varphi_{REF} - \varphi_{SAMPLE}} \right) \times \frac{\lambda_{THz}}{2\pi \times L_{s}}} + n_{o}}} & (31)\end{matrix}$

Note that it may be desirable to avoid the 2π phase ambiguity that mayresult if (φ_(REF)−φ_(SAMPLE))>2π; and as such, a constraint may beimplemented in which L_(s)×(n_(s)−n_(o))<λ_(THz). This may not be anissue for gas samples with indices near unity, but for solids andliquids, may be accomplished by using thin samples.

The imaginary part of the index of refraction of the sample, k (i.e.,the extinction coefficient) may be calculated as follows. From equation(20), and equations (29) and (30), the following relationship betweenthe reference and sample amplitudes E_(o) and E_(s) may be obtained:

$\begin{matrix}{^{{- 2}\pi \; k\frac{L_{s}}{\lambda_{THz}}} = \frac{E_{s}}{E_{o}}} & (32)\end{matrix}$

And from this expression, the extinction coefficient may be calculatedas follows;

$\begin{matrix}{k = {{- \frac{\lambda_{THz}}{2\pi \; L_{s}}}{\ln \left( \frac{E_{s}}{E_{o}} \right)}}} & (33)\end{matrix}$

Similar to before, the system 10 may scan through a number oftransmission frequencies in a range of frequencies, as shown in blocks54 and 56 of FIG. 3. As the path length rate scale factor S_(F) may beselected as a function of the transmission frequency (see equation(11)), the path length rate scale factor may be re-selected for eachtransmission frequency and may differ from one transmission frequency tothe next. For each transmission frequency in the range of frequency, andthus each beam of radiation having a different transmission frequency,the processor 38 may measure the amplitude and/or phase of thedown-converted signal current I_(Down). The resulting collection oftransmissions amplitudes and/or phases, and associated transmissionfrequencies, may define a measured absorption or dispersion signaturefor the sample in the sample cell 26, from which the sample may beidentified, as shown in block 58 of FIG. 3.

Many modifications and other embodiments of the invention will come tomind to one skilled in the art to which this invention pertains havingthe benefit of the teachings presented in the foregoing descriptions andthe associated drawings. Therefore, it is to be understood that theinvention is not to be limited to the specific embodiments disclosed andthat modifications and other embodiments are intended to be includedwithin the scope of the appended claims. Although specific terms areemployed herein, they are used in a generic and descriptive sense onlyand not for purposes of limitation.

1. A system comprising: a transmitter and receiver, the transmitterbeing configured to transmit an electromagnetic signal through a samplecell to the receiver at each of one or more selectable frequencies, thesample cell including a sample medium and a base medium, the receiverbeing configured to receive the electromagnetic signal and anotherelectromagnetic signal for mixing therewith, system including a firstpropagation path of the electromagnetic signal to the transmitter, and asecond propagation path of the other electromagnetic signal to thereceiver; an arrangement located along either of the first or secondpropagation paths, or along each of the first and second propagationpaths, the arrangement configured to alter the length of a respectivepropagation path such that the difference of the lengths of the firstand second propagation paths is altered at a pre-selected rate duringtransmission of the electromagnetic signal from the transmitter to thereceiver, and receipt of the electromagnetic signal and the otherelectromagnetic signal at the receiver; and a processor configured torecover an amplitude and phase of the transmitted electromagneticsignal, and configured to calculate a complex index of refraction of thesample medium as a function of the amplitude and phase of thetransmitted electromagnetic signal.
 2. The system of claim 1, whereinthe processor being configured to recover an amplitude and phase of thetransmitted electromagnetic signal includes being configured to receivea sequence of samples of the received electromagnetic signal, andDiscrete Fourier Transformation process the sequence of samples.
 3. Thesystem of claim 1, wherein the processor being configured to calculate acomplex index of refraction of the sample medium includes beingconfigured to calculate a real part and an imaginary part of the complexindex of refraction, the real part of the complex index of refractionbeing calculated as a function of the recovered phase of the transmittedelectromagnetic signal, and as a function of a recovered phase of anelectromagnetic signal transmitted through the sample cell including thebase medium but without the sample medium.
 4. The system of claim 3,wherein the processor being configured to calculate the real part of thecomplex index of refraction includes being configured to calculate thereal part n_(s) of the complex index of refraction according to thefollowing:$n_{s} = {{\left( {\varphi_{REF} - \varphi_{SAMPLE}} \right) \times \frac{\lambda}{2\pi \times L_{s}}} + n_{o}}$where φ_(SAMPLE) represents the recovered phase of the transmittedelectromagnetic signal, φ_(REF) represents the recovered phase of anelectromagnetic signal transmitted through the sample cell including thebase medium but without the sample medium, λ represents the wavelengthof the electromagnetic signal at a respective frequency, L_(s)represents a propagation path length through the sample medium, andn_(o) represents the free-space index of refraction.
 5. The system ofclaim 1, wherein the processor being configured to calculate a complexindex of refraction of the sample medium includes being configured tocalculate a real part and an imaginary part of the complex index ofrefraction, the imaginary part of the complex index of refraction beingcalculated as a function of the recovered amplitude of the transmittedelectromagnetic signal, and as a function of a recovered amplitude of anelectromagnetic signal transmitted through the sample cell including thebase medium but without the sample medium.
 6. The system of claim 5,wherein the processor being configured to calculate the imaginary partof the complex index of refraction includes being configured tocalculate the imaginary part k of the complex index of refractionaccording to the following:$k = {{- \frac{\lambda}{2\pi \; L_{s}}}{\ln \left( \frac{E_{s}}{E_{o}} \right)}}$where E_(s) represents the recovered amplitude of the transmittedelectromagnetic signal, E_(o) represents the recovered amplitude of anelectromagnetic signal transmitted through the sample cell including thebase medium but without the sample medium, λ represents the wavelengthof the electromagnetic signal at a respective frequency, and L_(s)represents a propagation path length through the sample medium.
 7. Thesystem of claim 1, wherein the arrangement comprises a pair ofarrangements each of which is located along a respective one of thefirst and second propagation paths, one of the arrangements beingconfigured to increase the length of one of the propagation paths, andthe other of the arrangements being configured to decrease the length ofthe other of the propagation paths, such that the difference of thelengths of the first and second propagation paths is increased at thepre-selected rate.
 8. The system of claim 1 further comprising: amodulator configured to modulate the electromagnetic signal transmittedby the transmitter, the modulator being configured to modulate theelectromagnetic signal at a frequency above the 1/f noise region of thereceiver.
 9. The system of claim 1, wherein the pre-selected ratecomprises a rate ω_(FS) selected to effectuate a path length modulationat a frequency: $\omega_{FS} = {\frac{2\pi}{\lambda}n_{F}S_{F}}$where λ represents the wavelength of the electromagnetic signal at arespective frequency, n_(F) represents the index of refraction of apropagating medium of the propagation paths, and S_(F) represents thepre-selected rate.
 10. The system of claim 9 further comprising: amodulator configured to modulate the electromagnetic signal transmittedby the transmitter, the modulator being configured to modulate theelectromagnetic signal at a frequency ω_(m), wherein the processor beingconfigured to recover an amplitude and phase of the transmittedelectromagnetic signal includes being configured to receive a sequenceof samples of the received electromagnetic signal, and Discrete FourierTransformation (DFT) process the sequence of samples at a firstfrequency of interest ω_(m)−ω_(FS), and DFT process the sequence ofsamples at a second frequency of interest ω_(m)+ω_(FS).
 11. A methodcomprising: selecting a rate of altering the difference of the lengthsof first and second propagation paths, the first propagation path beingof an electromagnetic signal to a transmitter configured to transmit theelectromagnetic signal through a sample cell to a receiver, the receiverbeing configured to receive the electromagnetic signal and anotherelectromagnetic signal for mixing therewith, and the second propagationpath being of the other electromagnetic signal to the receiver;transmitting the electromagnetic signal from the transmitter through asample cell to the receiver at each of one or more selectablefrequencies, and receiving the electromagnetic signal and the otherelectromagnetic signal at the receiver, the sample cell including asample medium and a base medium; altering the length of either or bothof the first or second propagation paths as the electromagnetic signalis transmitted from the transmitter to the receiver, and theelectromagnetic signal and the other electromagnetic signal are receivedat the receiver, either or both of the propagation paths being alteredsuch that the difference of the lengths of the first and secondpropagation paths is altered at the selected rate; and recovering anamplitude and phase of the transmitted electromagnetic signal, andcalculating a complex index of refraction of the sample medium as afunction of the amplitude and phase of the transmitted electromagneticsignal.
 12. The method of claim 11, wherein recovering an amplitude andphase of the transmitted electromagnetic signal includes receiving asequence of samples of the received electromagnetic signal, and DiscreteFourier Transformation processing the sequence of samples.
 13. Themethod of claim 11, wherein calculating a complex index of refraction ofthe sample medium includes calculating a real part and an imaginary partof the complex index of refraction, the real part of the complex indexof refraction being calculated as a function of the recovered phase ofthe transmitted electromagnetic signal, and as a function of a recoveredphase of an electromagnetic signal transmitted through the sample cellincluding the base medium but without the sample medium.
 14. The methodof claim 13, wherein calculating the real part of the complex index ofrefraction includes calculating the real part n, of the complex index ofrefraction according to the following:$n_{s} = {{\left( {\varphi_{REF} - \varphi_{SAMPLE}} \right) \times \frac{\lambda}{2\pi \times L_{s}}} + n_{o}}$where φ_(SAMPLE) represents the recovered phase of the transmittedelectromagnetic signal, φ_(REF) represents the recovered phase of anelectromagnetic signal transmitted through the sample cell including thebase medium but without the sample medium, λ represents the wavelengthof the electromagnetic signal at a respective frequency, L_(s)represents a propagation path length through the sample medium, andn_(o) represents the free-space index of refraction.
 15. The method ofclaim 11, wherein calculating a complex index of refraction of thesample medium includes calculating a real part and an imaginary part ofthe complex index of refraction, the imaginary part of the complex indexof refraction being calculated as a function of the recovered amplitudeof the transmitted electromagnetic signal, and as a function of arecovered amplitude of an electromagnetic signal transmitted through thesample cell including the base medium but without the sample medium. 16.The method of claim 15, wherein calculating the imaginary part of thecomplex index of refraction includes calculating the imaginary part k ofthe complex index of refraction according to the following:$k = {{- \frac{\lambda}{2\pi \; L_{s}}}{\ln \left( \frac{E_{s}}{E_{o}} \right)}}$where E_(s) represents the recovered amplitude of the transmittedelectromagnetic signal, E_(o) represents the recovered amplitude of anelectromagnetic signal transmitted through the sample cell including thebase medium but without the sample medium, A, represents the wavelengthof the electromagnetic signal at a respective frequency, and L_(s)represents a propagation path length through the sample medium.
 17. Themethod of claim 11, wherein altering the length comprises increasing thelength of one of the propagation paths, and decreasing the length of theother of the propagation paths, such that the difference of the lengthsof the first and second propagation paths is increased at the selectedrate.
 18. The method of claim 11 further comprising: modulating theelectromagnetic signal transmitted by the transmitter, theelectromagnetic signal being modulated at a frequency above the 1/fnoise region of the receiver.
 19. The method of claim 11, whereinselecting a rate comprises selecting a rate ω_(FS) to effectuate a pathlength modulation at a frequency:$\omega_{FS} = {\frac{2\pi}{\lambda}n_{F}S_{F}}$ where λ representsthe wavelength of the electromagnetic signal at a respective frequency,n_(F) represents the index of refraction of a propagating medium of thepropagation paths, and S_(F) represents the selected rate.
 20. Themethod of claim 19 further comprising: modulating the electromagneticsignal transmitted by the transmitter, the electromagnetic signal beingmodulated at a frequency ω_(m), wherein recovering an amplitude andphase of the transmitted electromagnetic signal includes receiving asequence of samples of the received electromagnetic signal, and DiscreteFourier Transformation (DFT) processing the sequence of samples at afirst frequency of interest ω_(m)−ω_(FS), and DFT processing thesequence of samples at a second frequency of interest ω_(m)+ω_(FS).